Causal Model Theory

Causal model theory (Waldmann, 1996, 2000, 2001; Waldmann & Hagmayer, 2001; Waldmann & Holyoak, 1992, 1997; Waldmann & Walker, 2005) emphasizes the importance of domain-independent knowledge and assumes that top-down and bottom-up processes interact in causal learning. While many studies have focused on the influence of domain-specific knowledge on causal induction, there is also abstract, domain-independent knowledge which might influence the acquisition and use of causal knowledge.

An example of domain-independent knowledge is the fact that all causal relations are inherently asymmetric: causes generate effects but not vice versa. In the spirit of Kant (1781/1974) it is assumed that our experiences are constrained by and interact with this abstract, domain-independent piece of knowledge. With reference to causal learning it is postulated that abstract causal knowledge provides the background against which we evaluate covariational information. For example, in a standard causal learning paradigm participants are provided with events precategorized as cause and effect (e.g., fertilizer and blooming). Thus, before learners are confronted with any covariational data, they already have a qualitative understanding of how the events are related to each other: a representation implying causal directionality. Data-driven accounts lack the representational power to express this directionality. For example, associative theories use cue and outcome as the two basic types of event representations with the cue defined as the event that triggers the outcome irrespective of their actual causal roles.

Contrary to covariational accounts, causal model theory (henceforth CMT) also explicitly differentiates between causal strength and causal structure. The tacit assumption of covariational approaches is that statistical information not only allows for estimates of causal strength but simultaneously provides information about the underlying causal structure (i.e., the absence of a causal relation is considered as a special case of zero causal strength). Indeed, this idea is plausible in experiments characterized by single or multiple causes which are directly related to the effect(s).

However, in both everyday and scientific causal learning we are often confronted with complex causal networks consisting of several variables. Figure 2 shows three fundamental causal structures: a common-effect model in which event X is generated (independently or jointly) by both Y and Z, a common-cause model in which X is a cause of both Y and Z, and a causal chain in which X causes Y which, in turn, causes Z.

In principle, by combining these basic structures causal models can be constructed for

causal relations of any level of complexity. Such causal models provide a qualitative representation of causal systems, that is, they only state the (hypothesized) existence of certain causal relations without specifying their strength. This idea

fits with the intuition that we often have a rather qualitative understanding of causal relations without exact knowledge of their strength. For example, we might know that greenhouse gases affect the climate or that we might get fat from eating too much fast food without knowing the exact strength of these relations. Therefore some authors have argued for a priority of structure over strength (Griffiths & Tenenbaum, 2005; Lagnado, Waldmann, Hagmayer, & Sloman, in press; Pearl, 2000; Waldmann, 1996).

According to CMT statistical information is not treated as context-free input to the process of causal induction but is evaluated with reference to a hypothesized causal structure. Thus, covariational knowledge serves the purpose of helping to estimate a causal model’s parameters. Note that causal directionality is a necessary prerequisite for representing structured models which convey more information than simply stating that certain events are correlated. For example, both the common-cause structure (Figure 2b) and the causal chain (Figure 2c) imply that events Y and Z are correlated. However, encoding causal knowledge in form of causal models also conveys the information that in the common-cause model this relation is spurious. Moreover, different causal models have statistical implications which learners can use to evaluate regularity information and decide between alternative causal models. For example, in a common effect model with independent causes (Figure 2a), events Y and Z become dependent conditional on values of their common effect X. In contrast, the common-cause model implies that Y and Z become independent conditional on values of their common cause X (“so-called

“explaining away” effect, in the psychological literature also known as discounting principle). Constraint-based methods of causal induction capitalize on these statistical implications to reveal causal structure from statistical information (cf. Section 4.2).

Several studies demonstrate how abstract causal knowledge influences causal learning. For example, it has been shown that causal models mediate cue interaction effects such as blocking and overshadowing. While associative accounts predict cue competition regardless of the cues’ causal roles, CMT predicts cue competition only for

(a) (b) (c)

Figure 2. Basic causal models. a) Common-effect model (CE) b) Common-cause model (CC) c) Causal chain (CH)

PSYCHOLOGICAL M_{ODELS OF} C_AUSAL C_OGNITION 32 causes but not for effect events, a prediction confirmed in a number of studies

(Waldmann, 2000, 2001; Waldmann & Holyoak, 1992; Waldmann & Walker, 2005).

However, there is also evidence that the effect is sometimes influenced by associative processes (Allan & Tangen, 2005; Cobos et al., 2002; Tangen & Allan, 2004; Tangen et al., 2005). It has also been shown that learners use causal models to select the events they want to control for to give unconfounded estimates of causal strength. As discussed in Section 3.2.2 it is not always appropriate to conditionalize on all causally relevant events. For example, in the common-effect structure depicted in Figure 2a it is normatively correct to conditionalize on the absence of Y to estimate the causal strength of the link Z→X. However, if the three events form a causal chain X→Y→Z, conditionalizing on the intermediate event Y renders X and Z independent thus erroneously indicating that the two events are not causally related. CMT predicts that learners select the variables to control for in accordance with their assumptions about the underlying causal structure. In a series of experiments, Waldmann and Hagmayer (2001) tested these predictions and demonstrated that manipulations of the suggested causal structure yield very different causal judgments derived from identical covariational information. There is also evidence that learners can integrate separately learned causal relations to more complex causal structures to predict unobserved covariations (Hagmayer & Waldmann, 2000). In addition to causal learning, CMT has also successfully been applied to categorization (Rehder, 2003a, 2003b; Rehder &

Burnett, 2005; Rehder & Hastie, 2001; Waldmann, Holyoak, & Fratianne, 1995). These studies show that learners do not simply use correlated features to classify objects but take into account the internal causal structure of the entities to determine their category membership.

The main challenge for CMT is, of course, to explain how we acquire hypotheses about causal structure in the first place. One possible answer is that we use non-statistical cues such as temporal order (Lagnado & Sloman, 2004, in press), or interventions (Hagmayer, Sloman, Lagnado, & Waldmann, in press; Woodward, 2003), or that we generate hypotheses by analogy (cf. Holyoak & Thagard, 1995). Recently, algorithms have been developed which aim to uncover causal structure by analyzing the conditional dependence and independence relations found to hold in the data (Glymour

& Cooper, 1999; Pearl, 2000; Spirtes et al., 1993) or use Bayesian methods to compute the likelihood of the data given a causal model (Steyvers et al., 2003). These methods are discussed in detail in Section 4.2.